Some engineers might still worry that a coefficient function like the sinc function must be inherently evil or suspect, just because of its pattern, that rings and has 'pre-response' before its highest peak, which thus must be 'unnatural' and 'non-causal', like a bell ringing before it is struck. But they are doubly wrong.
First, when exactly does the causal bell strike occur? It is at the very left beginning of the coefficient function that embodies the design of the digital filter. The highest peak does occur later in time, but that's because all filters operate by introducing latency or delay, whereby their output of a given transient peak occurs later in time than that same transient peak appeared at the filter's input. The sinc brickwall filter, like all filters, is purely causal, since the initiating bell strike occurs at the very beginning left end, and the peak transient is simply output at a later time.
Second, look at the sinc function's pattern and profile, that some engineers find so 'unnatural' and 'non-causal': a long 'pre-ring' that gradually builds up to a peak amplitude, followed by a gradually declining post-ring. Engineers also believe that this pattern is the 'impulse response' of a sinc brickwall filter, i.e. its response to a pulse of short duration, which is why they shun it.
Now, consider a pulse response W that looks almost identical to the sinc 'pulse response', and has exactly the same key pattern and profile features that engineers find so 'unnatural' and 'non-causal' in the sinc function (ringing, 'pre-response', bell strike peak in middle instead of causally at beginning, slowly rising and falling long duration ringing). This pulse response W would of course also be pronounced to be 'unnatural' and 'non-causal', indeed as something that could not possibly occur in nature. But these engineers are very wrong.
This pulse response W, having exactly the same pattern features as the sinc pulse response, is produced by every musical wind instrument, from a piccolo to an organ pipe, and even by the caveman's very first musical instrument, a hollow tube. Last time I checked, musical wind instruments do operate in accordance with the natural and causal laws of physics. The pulse signal input is simply a rectangular pulse of DC air pressure applied for a period of time, that starts at the very left beginning of the wind instrument's ringing air resonance gradual buildup, and stops the instant that this building resonance first reaches its peak value, thus producing just one highest peak, followed by a gradual decline of the ringing resonance. So, are all musical wind instruments 'unnatural' 'non-causal' agents of the devil that must be shunned, just like the sinc brickwall filter?
We now know, from our discussion above, that the sinc brickwall filter has the ideal, and only perfect, coefficient function for correctly passively gathering information from a very wide field of data points, to correctly reconstruct any signal waveform, and thus also exhibits perfect time domain transient response performance.
We also now know that, if some engineer makes the blunder of thinking that the sinc coefficient function for passively gathering information also represents the sinc brickwall filter's actual impulse response for actively radiating energy, he will mistakenly pronounce the sinc filter's impulse response and hence time domain transient response as awful, so he will completely reject the sinc brickwall filter and join the modern revisionist movement designing alternative 'time domain optimized' filters (that in fact will necessarily butcher the signal waveform with truly awful time domain distortions).
And we will soon know (see below) that the engineers worldwide - who mistakenly believe the digital impulse response test is informative and evaluatively valid, and who don't realize that this so-called impulse response 'test' is not a filter performance test at all, but instead merely photocopies the filter coefficient function - will be fooled into the totally erroneous (indeed backwards) belief that their so-called impulse response test proves the active energy radiating performance of the filter is as bad as the coefficient function seems to look, or as good as it seems to look.
A further crucial distinction. The filter's coefficient function is actually one input to the filter's convolution process, a process that combines two distinct inputs (the other is the signal waveform) to produce the filter's output. In contrast, the filter's time domain transient response performance appears at the very opposite filter port, the output. Engineers should know better than to conflate the input to a DUT with the output from a DUT. To see this clearly, it's better to regard a filter DUT as merely a convolution engine black box, that convolves the information it passively gathers from 2 input functions (signal and coefficient function), to statically calculate each dot of output function.
In truth and point of fact, the coefficient function input to this filter's convolution process can look as weird as it pleases, and we have no basis or right for passing judgment on it. Nor can we pass judgment on a mere photocopy of the appearance of this coefficient function input (as discussed below, a mere photocopy is all that the so-called digital 'impulse response test' does). The only thing that counts is that the output from the filter accurately reconstructs the original signal waveform.
Here, at the filter output port, is where the sinc brickwall filter design, however ugly its coefficient function might be, actually delivers the correct accurate signal waveform. Here, at the filter output port, is where the modern revisionist 'time domain optimized' filter designs, however beautiful their coefficient functions might be, actually deliver horribly butchered time domain distortions of the signal waveform, with MQA being worst of all.
This is yet further proof that the revisionist engineer, who pays attention only to the beautiful appearance of his input coefficient function (and to its photocopy image by the invalid so-called 'impulse response test'), is so uncomprehending of how digital actually works, that for some odd reason he never pays attention either to looking at the actual output from his digital filter to notice how distorted its time domain reconstruction is, or to learning how the convolution process (the essential core of every filter) actually works internally, and thus understanding how and why his beautiful looking coefficient filter input actually produces such an ugly duckling time domain distortion in its output from his reconstruction filter.
Second, the digital version of the impulse response test does not even qualify as being a test of time domain performance, neither of a digital filter design nor of anything else. Moreover, it actually does not even scientifically qualify as being a test at all in the first place.
Why? Briefly, the digital version of the impulse response test, far from testing the actual performance of a digital filter design embodied in the coefficient function that the digital design engineer first sketched on a napkin at dinner (or testing anything at all), actually merely photocopies the engineer's own sketch on that napkin. Digital engineers worldwide pretend (and erroneously believe) that the filter designer can test the actual performance of his filter design, and its actual performance specifically in time domain transient response, by merely holding his very own napkin sketch up to a mirror and merely looking at it, or by merely looking at his own napkin sketch itself.
The digital impulse response test fails to test any actual filter performance because it never activates the convolution process that constitutes the digital filter's performing process. It feeds only a single non-zero number (as the test signal) into this convolution process, but convolution depends on mathematical addition to act, and (as all of us learned at age 5) addition requires at least 2 numbers to add. Note that the zeroes on both sides of the single number don't count here, because (as all of us learned at age 6), adding zero to anything is a null operation, mathematically equivalent to never having done an adding operation at all.
Furthermore, the digital impulse response test further fails to test any actual filter performance because that single test signal number it inputs is mathematically a 1. Convolution also depends on mathematical multiplication to act. As all of us learned at age 6, multiplication by 1 is also a null operation, returning merely a photocopy of the entity being multiplied by 1 (here the napkin sketch of the engineer's own coefficient function design).
As all of us learned at age 4 from Goldilocks, the only way to test the actual performance of a bed's comfort is to make it actually perform the relevant design task (i.e. the task that the device was designed to perform), by actually lying down on the bed. And so, even as an age 4 child, all of us already knew what brilliant PhD digital engineers worldwide have failed for 34 years to comprehend about how digital actually works, and about how the impulse response test works.
Thus, the only way to test the impulse response of a digital filter is to make it actually perform the convolution process. This mandates that the input test signal be a shaped (brief) pulse, comprising 2 or more non-zero numbers, not the single sample impulse (aka Kronecker delta) that has been universally used to date. It is simply wrong to mindlessly, willy-nilly pursue the supposed conceptual ideal of employing the briefest possible impulse as a test signal, without comprehending how digital actually works, and without using this knowledge to make the digital filter (like Goldilocks) actually perform a relevant convolution task, in order to test its performance.
* * * * * *
Consider the following mattress design parable. A grown-up Golidlocks now runs her own mattress factory, and her mattress design is a conventional innerspring mattress, employing hundreds of helical coil springs that are softly interlinked by twine. This mattress design actually does provide superb comfort, when actually performing its design task of supporting a human body. One day, Baby Bear, fresh out of mattress design university with his PhD in hand, comes to visit Goldilocks.
Baby Bear says to Goldilocks: "Look at my drawing sketch of your mattress design. Your old fashioned design is badly flawed, in several ways. Look at the hundreds of peaks and valleys among the hundreds of coiled springs, constituting a complex ringing pattern of hundreds of reactive compliances, that also make matters more complex by interfering with one another. This complex and far reaching ringing array of springs provides a very confused and complex support, and thus cannot possibly be comfortable. Furthermore, your mattress design is flawed by violating temporally unidirectional causality, because your helical springs are wound counter-clockwise, whereas everyone knows that time itself, hence temporal causality, proceeds unidirectionally clockwise. Moreover, your mattress design is flawed because it exhibits unnatural non-causal pre-response; when you move on the mattress, say by rolling over, some of your linked coil springs start compressing in advance, before your moving body even touches them (or that part of the mattress that they support)."
Baby Bear continues: "Now, in contrast, look at my drawing sketch of my new revisionist mattress design. You'll notice that the entire human body is supported by one single large nail spike pointing upward, in the center of the mattress. My brand new revisionist mattress design cures all the flaws in your old fashioned design, providing a pure single point support, with no ringing reactive compliance flaws, and no complex ringing interactions, and no unnatural non-causal pre-response. My single-spike mattress design provides much faster and more direct time domain comfort response - without the long reactive ringing, and also complex interference behavior, of your old fashioned mattress design. The customer decisively knows if he's lying on or off the single sharp supporting spike. And my single-spike design has superb time domain performance, with a virtually perfect instant impulse response, since the customer gets instant comfort response the moment he gets onto the sharp spike, and also the moment he gets off it."
Goldilocks replies: "Yes, but does this 'ideal' time domain impulse response mattress design of yours actually have any ability to actually perform its design task, of providing comfort to the customer's body? What good is 'perfect' impulse response if your device is actually horrible at performing its design task, or worse yet incapable of performing it at all? It's obvious that you don't have a clue, about comprehending how mattresses actually work, in actually performing their design task of supporting a human body comfortably."
Goldilocks continues: "Nor indeed do you even comprehend how science and engineering actually work. The way that my design's complex array of many coil springs (softly interconnected) actually does work, and does superbly accomplish its design task of supporting the entire human body comfortably. For example, some parts of the human body are heavier and/or protrude more, so these body parts need to each be individually supported with their own custom portion of the complex coil spring array, for optimum comfort. For another example, the soft interconnection of the coil springs in fact provides actual performance that, via pre-response, makes the mattress more comfortable, so that, when you roll over, the next coil spring you get to has anticipated your arrival and is already pre-kneeling to provide a gradual curve-fitted path for your body, rather than being at full height where your body would bump into it as an abrupt sharp corner."
"Also, you are spouting anti-scientific gibberish when you assert that the counterclockwise direction of the coil springs makes my mattress design 'unnatural' and 'non-causal' - and for several reasons: time itself does not move clockwise nor counterclockwise, since this a pure convention as to how analog clocks portray the passage of time; the coil spring itself might look counterclockwise from one end, but then it is clockwise viewed from its other end; and the helical spring coils, when actually performing their design task, move only vertically, not helically, so there is not even any motion in either a clockwise or counterclockwise direction, hence there cannot possibly be any violation of temporal causality in one helical direction vs. the other."
"Now, as to your new revisionist single-spike mattress design, it's obvious again that you don't have a clue about how mattresses actually perform their design task of supporting the complete human body, with optimized comfort. You have created your revisionist design so that its portrayal in your own drawing sketch looks subjectively pretty to your eye, with its elegantly clean and simple single spike. But clearly you haven't even begun to think about how well or badly it might actually perform its design task of supporting a whole human body in comfort. Moreover, if you had the slightest clue about comprehending how mattresses actually perform their design task, you would have instantly realized that your revisionist single-spike design is in fact horribly uncomfortable, actually providing the worst possible comfort of any design, and indeed it even utterly fails to support all parts of the human body."
"Let me help you out. Go down that hallway and talk to my factory foreman, and show him your drawing sketch of your revisionist single-spike mattress design. Ask him to build you a physical real prototype. Then lay yourself down on that prototype, to conduct a comfort response test, which makes your device under test actually perform its design task of supporting your entire body with comfort."
So Baby Bear starts down that hallway, with both of his drawing sketches in hand: his sketch of his own new single-spike design, and his sketch of Goldilocks' old fashioned multiple spring design. But, before he gets to the factory foreman, he sees a photocopy machine room on the left side. So he goes in there, and he makes photocopies of both drawing sketches. He then returns to Goldilocks, with the two photocopies in hand, and gives these two photocopies to Goldilocks.
Baby Bear says: "I just conducted that comfort response test on both mattress designs. Here you see the test results. Just compare the two pictures here, showing the comfort response test results, conducted on both alternative mattress designs. The test result portrait from my new revisionist design shows and proves, by this comfort response test measurement, that it has much cleaner, purer, quicker, simpler direct comfort response - without all that messy complex behavior that your old fashioned mattress design evinces in its test result portrait from this same comfort response test."
Goldilocks shakes her head. "You fool! You have absurdly blundered in so many ways, and so severely! You never even tried to obey my nursery tale lesson, that to evaluate any device you need to make that device actually perform its design task. You never even built that prototype of your single-spike design, in order to be able to test its actual performance. Furthermore, these two photocopies don't represent the output response from any comfort response test measurement. At best, these design drawing sketches can represent only the input data to a comfort response test measurement, not the output from such a test measurement. By merely photocopying the input to a test measurement, you in fact have merely taken the input (to such a comfort response test), and you have merely multiplied it by 1, thus merely duplicating your own drawing sketch of your own design - and therefore not even conducting any test whatsoever."
"Since you in fact have not conducted any test at all, and also since you obviously don't comprehend at all how mattress designs actually work in their design task of supporting all parts of the human body in comfort, everything you are doing, including your entire engineering approach and programme, is not only unscientific, but also damagingly anti-scientific and anti-engineering. Just consider what you are actually doing here. Instead of testing the actual performance of your design, you are merely taking your own drawing sketch of your design concept, and multiplying it by 1 (photocopying it), which is equivalent to you holding your own drawn sketch of your own design up to a mirror, and then 'evaluating' the purported actual performance of your own design by its appearance reflected from that mirror. Then, since you obviously don't comprehend how mattresses actually work, your purported 'evaluation' of your own sketch via that mirror reflection actually objectively reduces to whether you subjectively opine that the image you see is "pretty". From an objective outside viewpoint, all you are actually doing is holding up your sketch of your own design concept, and subjectively judging it to be pretty, i.e. subjectively pleasing to your eye (likewise, when you hold up your sketch of the Goldilocks multiple coil mattress design, you are in objective fact merely subjectively judging it to be ugly, since you don't comprehend at all how this multiple coil design actually works, and actually performs its design task superbly)."